Spin effect on the low-temperature resistivity maximum in a strongly interacting 2D electron system

The increase in the resistivity with decreasing temperature followed by a drop by more than one order of magnitude is observed on the metallic side near the zero-magnetic-field metal-insulator transition in a strongly interacting two-dimensional electron system in ultra-clean SiGe/Si/SiGe quantum wells. We find that the temperature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\text {max}}$$\end{document}Tmax, at which the resistivity exhibits a maximum, is close to the renormalized Fermi temperature. However, rather than increasing along with the Fermi temperature, the value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\text {max}}$$\end{document}Tmax decreases appreciably for spinless electrons in spin-polarizing (parallel) magnetic fields. The observed behaviour of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\text {max}}$$\end{document}Tmax cannot be described by existing theories. The results indicate the spin-related origin of the effect.

The existence of the metallic state in strongly interacting 2D electron systems is intimately related to the existence of spin and valley degrees of freedom [34][35][36][37] . If the electron spins become completely polarized by a magnetic field B * parallel to the 2D plane, the spin degeneracy that determines the Fermi energy changes to g s = 1 , corresponding to spinless electrons. In a thin 2D electron system in the metallic regime, the resistivity increases with parallel magnetic field by a factor of a few and saturates above the polarization field B * 38-45 . (An attempt was made by M. S. Hossain et al. 46 to analyze the data for B * in the insulating phase. However, such data reflect the physics of localized electron moments in the band tail, which is entirely different from that of the metallic phase (see, e.g., Ref. 9 ), and, therefore, the conclusion of M. S. Hossain et al. on the ferromagnetic state is not justified.) The metallic temperature dependence of the resistivity is suppressed in the spin-polarized regime, as observed in silicon metal-oxide-semiconductor field-effect transistors (MOSFETs) 38,43,47 , p-type GaAs/AlGaAs heterostructures 42,45 , and narrow AlAs quantum wells 44 . Recent work has established that the ultra-clean 2D electron system in SiGe/Si/SiGe quantum wells, which is similar to the clean Si MOSFETs in that the 2D electrons host in Si but is distinguished mainly by the much higher electron mobility, still exhibits a metal-insulator transition even at B = B * that is determined using a number of different measurement methods 9,43 and is attributed to the existence of two distinct valleys in its spectrum 48 .
In this paper, we report studies of the non-monotonic temperature dependence of the resistivity on the metallic side near the metal-insulator transition in a strongly interacting, spin-unpolarized ( g s = 2 ) as well as fully spin-polarized (or spinless, g s = 1 ) bi-valley 2D electron system in ultra-clean SiGe/Si/SiGe quantum wells. We find that in zero magnetic field, the temperature T max , at which the resistivity has a maximum, is close to the renormalized Fermi temperature T F , which is in agreement with the dynamical mean-field theory. However, rather than increasing along with the Fermi temperature, the value T max decreases appreciably for spinless electrons in spin-polarizing magnetic fields, which is in contradiction with this theory. A scaling analysis of ρ(T) in the spinless electron system in the spirit of DMFT shows that the low-temperature resistivity drop is still described by the theory, similar to the case of the spin-unpolarized electron system. At the same time, the reduced value of T max in spin-polarizing magnetic fields is consistent with the predictions of the renormalization-group scaling theory, but T max in zero magnetic field is in disagreement with this theory. Thus, the observed behaviour of T max cannot be described by existing theories. Nor can it be explained in terms of the increase of the residual disorder potential and the reduction of the electron interaction strength due to the reduced spin degrees of freedom in spin-polarizing magnetic fields, because the relation T max ∼ T F still holds for clean Si MOSFETs 28,29 and low-mobility Si/SiGe quantum wells 20 in zero magnetic field. This indicates the spin-related origin of the effect.
Raw data for the resistivity as a function of temperature are shown at zero magnetic field ( g s = 2 ) in Fig. 1a and at B = B * ( g s = 1 ) in Fig. 1b for electron densities above the critical electron densities for the metal-insulator transition n c (0) ≈ 0.88 × 10 10 cm −2 and n c (B * ) ≈ 1.1 × 10 10 cm −2 , respectively. In zero magnetic field, the ρ(T) curves are non-monotonic with the maxima at density-dependent temperatures T max (0) over a wide range of electron densities n s ; below T max (0) , the resistivity drops sharply with decreasing temperature so that the drop can exceed an order of magnitude (see the inset to Fig. 1a). At B = B * , the ρ(T) curves are non-monotonic in a narrower range of electron densities, and the resistivity drop below T max (B * ) weakens. The values of B * are density-dependent and have been determined by the saturation of the ρ(B) dependences (see the inset to Fig. 1b), which corresponds to the lifting of the spin degeneracy 40,41 . Magnetic fields used in our experiments were within the range between approximately 1 and 2 T. The measurements were restricted to 0.5 K because this was the www.nature.com/scientificreports/ highest temperature at which the complete spin polarization could still be achieved; the restriction is likely to reflect the degeneracy condition for the dilute electron system with low Fermi energy. In Fig. 2, which is the main figure of this paper, we plot the values of T max as a function of the electron density in B = 0 and B = B * . The data for T max (B * ) lie significantly lower than those for T max (0) . Interestingly, each dependence can be described by a linear function that extrapolates to zero at n s close to n c (0) or n c (B * ) , and the slopes of both dependences are close to each other. We also plot the calculated values of renormalized Fermi temperatures T F for both cases. In zero magnetic field, the density dependences of the resistivity maximum temperature T max (0) and the Fermi temperature T F (0) are close to each other in the electron density range where they overlap. However, there is a qualitative difference between the behaviour of T max and that of T F when lifting the spin degeneracy. Rather than increasing along with the Fermi temperature, the value T max decreases when polarizing electron spins.
The Fermi temperature T F (B * ) has been calculated from the low-temperature value B * (see the inset to where k B is the Boltzmann constant, g v = 2 is the valley degeneracy, m is the renormalized energy-averaged effective mass that is determined by the density of states, g F ≈ g 0 = 2 is the g-factor at the Fermi level, g 0 is the g-factor in bulk silicon, and µ B is the Bohr magneton. We argue that the Fermi temperature T F (0) of spinunpolarized electrons is approximately half of the Fermi temperature T F (B * ) of completely spin-polarized ones. Indeed, it was experimentally shown in Ref. 49 that the electron spin magnetization is proportional to the parallel magnetic field in the range up to B = B * for the clean, strongly interacting 2D electron system in Si MOSFETs that is similar to the 2D electron system in SiGe/Si/SiGe quantum wells. (For strongly disordered Si MOSFETs, the band tail of localized electrons persists into the metallic regime 50 in which case both the nonlinear magnetization as a function of parallel magnetic field and the shift of the dependence B * (n s ) to higher densities are observed due to the presence of localized electron moments in the band tail 9,51-54 .) Taking into account the smallness of the exchange effects in the 2D electron system in silicon so that the g-factor is approximately constant close to g 0 = 2 at low densities 8,9,24 , this indicates that the renormalized density of states in a spin subband is approximately constant below the Fermi level, independent of the magnetic field. Therefore, the change of T F when lifting the spin degeneracy should be controlled by the change of g s . As concerns the band flattening corresponding to a peak in the density of states at the Fermi level, observed in the 2D electron system in SiGe/Si/SiGe quantum wells, the Fermi energy is not particularly sensitive to this flattening, at least, not too close to the critical point 24 . So, one expects that the relation T F (0) ≈ T F (B * )/2 holds for the data in question. We stress that its accuracy is not crucial for our qualitative results.
In Fig. 3, we plot the ratio δρ/δρ max = (ρ(T) − ρ(0))/(ρ(T max ) − ρ(0)) as a function of T/T max in B = B * so as to check the applicability of the DMFT. The curve for the highest electron density follows the theoretical dependence in the weak-disorder limit at all temperatures. Two other curves for lower electron densities also follow the theoretical dependence at T ≤ T max but deviate from the theory at higher temperatures, revealing the behaviour similar to that observed at low n s in zero magnetic field 30 . Albeit the density range of the applicability of DMFT to the completely spin-polarized system is not as wide as that in B = 0 , the low-temperature resistivity drop is described by the theory, similar to the case of the spin-unpolarized electron system. For completeness, in the inset to Fig. 3 we plot the ratio ρ/ρ max in the fully spin-polarized system as a function of www.nature.com/scientificreports/ ρ max (e 2 /πh) ln(T/T max ) , which is the scaling form suggested by the renormalization-group scaling theory 34,35 .
The data do not scale in the range of electron densities studied. The dynamical mean-field theory successfully describes the closeness of T max and the renormalized Fermi temperature T F in zero magnetic field, as well as the resistivity drop at temperatures below T max in both spinunpolarized and fully spin-polarized electron systems. However, the observed decrease of T max when lifting the spin degeneracy is opposite to the predictions of DMFT. At the same time, the reduced value of T max in spin-polarizing magnetic fields is consistent with the predictions of the renormalization-group scaling theory, but T max in zero magnetic field is in disagreement with this theory. The observed behaviour of T max cannot be described by existing theories.
In view of the competition between electron-electron interactions and disorder, one can expect the value T max to decrease with increasing disorder level or decreasing electron interaction strength, as occurs in our case when lifting the spin degeneracy. However, the increase of the residual disorder potential when lifting the spin degeneracy in SiGe/Si/SiGe quantum wells, manifested in the metallic regime by the resistivity increase by a factor of a few (the inset to Fig. 1b), cannot be the origin for the observed weakening of the resistivity drop at temperatures below T max and shift of the maximum in ρ(T) to lower temperatures. Indeed, we compare to the electron system of clean Si MOSFETs, which is different from the studied SiGe/Si/SiGe electron system mainly by lower electron mobility, whereas at the B = 0 metal-insulator transition in both systems, the values of the interaction parameter are similar, r s ≈ 20 (for details, see Ref. 12 ); the interaction parameter is defined as the ratio between the Coulomb and (bare) Fermi energies, r s = g s g v /2(πn s ) 1/2 a B (where a B is the effective Bohr radius in semiconductor). Importantly, both systems correspond to the clean limit where the electron interactions are dominant over disorder effects, which is the regime we are interested in. In this case, the zero-magnetic-field metal-insulator transition occurs close to the electron density at which the effective mass at the Fermi level tends to diverge 12 . In clean Si MOSFETs, where the electron mobility is some two orders of magnitude lower than that in SiGe/Si/SiGe quantum wells studied here, the resistivity drop at T < T max in zero magnetic field reaches a factor of 7, which is comparable to that in our samples. Also, in clean Si MOSFETs, the positions of the ρ(T) maxima in B = 0 closely follow the renormalized Fermi temperatures 28,29 , which is similar to our finding in B = 0 . Neither can the reduction of the electron interaction strength due to the reduced spin degrees of freedom in spinpolarizing magnetic fields be the origin of the observed behaviour of T max . Indeed, the relation T max ∼ T F still holds for low-mobility Si/SiGe quantum wells in zero magnetic field at electron densities above n s ≈ 10 11 cm −2 20 , which corresponds to the interaction parameter smaller by a factor of a few compared to that in our range of electron densities. This indicates the spin-related origin of the behaviour of T max observed in our samples.
It is worth noting that the suppression of the metallic regime and the increase of the transition point when lifting the spin degeneracy in a strongly correlated 2D electron system (see, e.g., Fig. 2, as well as earlier experimental data 43,[55][56][57] ) are explained taking account of the spin and valley degrees of freedom, according to theories [34][35][36][37]58,59 . Similarly, within the DMFT, the critical density is predicted to increase in spin-polarizing magnetic fields 26 .
In conclusion, we have studied the non-monotonic temperature dependence of the resistivity on the metallic side near the metal-insulator transition in a strongly interacting, spin-unpolarized and spinless two-valley 2D electron system in ultra-clean SiGe/Si/SiGe quantum wells. We have found that in zero magnetic field, the resistivity maximum temperature T max is close to the renormalized Fermi temperature T F , which is in agreement with the dynamical mean-field theory. However, rather than increasing along with the Fermi temperature, the value T max decreases appreciably for spinless electrons in spin-polarizing magnetic fields, which is in contradiction with the theory. The DMFT quantitatively describes the low-temperature resistivity drop in both spin-unpolarized and spinless electron systems. At the same time, the reduced value of T max in spin-polarizing magnetic fields is consistent with the predictions of the renormalization-group scaling theory, but T max in zero magnetic field is in disagreement with this theory. Thus, the observed behaviour of T max cannot be described by existing theories. Nor can it be explained in terms of the increase of the residual disorder potential and the reduction of the electron  26,28,29 . The inset shows the analysis based on the scaling form suggested by the renormalization-group scaling theory 34 www.nature.com/scientificreports/ interaction strength due to the reduced spin degrees of freedom in spin-polarizing magnetic fields, because the relation T max ∼ T F still holds for clean Si MOSFETs and low-mobility Si/SiGe quantum wells in zero magnetic field, indicating the spin-related origin of the effect. To describe the observed behaviour of T max within a single approach that treats properly the spin effects on the low-temperature resistivity in a strongly interacting 2D electron system in parallel magnetic fields, further theoretical efforts are necessary.

Methods
The samples we used were ultra-high mobility SiGe/Si/SiGe quantum wells. The peak electron mobility in these samples reaches 240 m 2 /Vs. The approximately 15 nm wide silicon (001) quantum well was sandwiched between Si 0.8 Ge 0.2 potential barriers. The samples were patterned in Hall-bar shapes with the distance between the potential probes of 150 μm and width of 50 μm using standard photo-lithography. The long side of the Hall bar corresponded to the direction of current parallel to the [110] or [ 1 10] crystallographic axis. The in-plane magnetic field was perpendicular to the current to exclude the influence of ridges on the quantum well surface on the resistance measured at high electron densities (for more details, see Refs. 60,61 ). Measurements were carried out in an Oxford TLM-400 dilution refrigerator with a base temperature of ≈ 25 mK. Data were taken by a standard four-terminal lock-in technique in a frequency range 0.5-11 Hz in the linear regime of response.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.